How simple geometry gives us the formula for a chord

When we approximate the length of a curved arc, we often divide the arc into many small pieces. Each curved piece can then be approximated by a straight line joining its two endpoints.

That straight line is called a chord.

It is important to keep the notation clear:

$s$ represents arc length.
$\Delta s$ represents a finite change in arc length.
$c$ will represent the chord length.

Thus, we will derive a formula for $c$ , not $\Delta s$ .

Q1. What is a chord?

Consider a circle with center $O$ .

Choose two points $A$ and $B$ on the circle.

The straight line joining $A$ and $B$ is the chord.

The curved part of the circle between $A$ and $B$ is the corresponding arc.

The chord is straight, whereas the arc is curved.

For a finite arc,

$\boxed{c<\Delta s}$

because the straight path is shorter than the curved path.

Q2. What information do we need to find the chord?

Suppose the circle has radius

$R$

and the angle subtended by the chord at the center is

$\Delta\theta$

So we have:

$OA=R$
$OB=R$
$\angle AOB=\Delta\theta$
$AB=c$

We want to find $c$ .

Q3. What kind of triangle do we have?

The two radii have the same length:

$OA=OB=R$

Therefore, triangle $OAB$ is an isosceles triangle.

Now draw a line from the center $O$ to the midpoint $M$ of the chord $AB$ .

Because the triangle is isosceles, $OM$ has two important properties.

It:

divides the chord into two equal parts;
divides the central angle into two equal parts.

Therefore,

$AM=MB=c/2$

and

$\angle AOM=\Delta\theta/2$

Q4. What does the right triangle $OAM$ look like?

Now we can focus only on triangle $OAM$ .

The important quantities are:

$OA=R$ $AM=c/2$

and

$\angle AOM=\Delta\theta/2$

Since $OM$ is perpendicular to $AB$ , this is a right triangle.

Q5. Why do we use sine?

Recall the definition of sine in a right triangle:

$\sin(\theta)=\text{opposite side}/\text{hypotenuse}$

For our triangle, the angle is

$\Delta\theta/2$

The side opposite this angle is

$AM=c/2$

The hypotenuse is

$OA=R$

Therefore,

$\sin(\Delta\theta/2)=(c/2)/R$

Q6. How do we solve this for the chord length?

Starting with

$\sin(\Delta\theta/2)=c/(2R)$

multiply both sides by $2R$ :

$\boxed{c=2R\sin(\Delta\theta/2)}$

And that is the chord length formula.

Notice that it came entirely from:

$\boxed{\text{isosceles triangle}+\text{right triangle}+\text{sine}}$

No calculus was required.

Q7. Why is there a $\Delta\theta/2$ rather than $\Delta\theta$ ?

This is an important point.

Originally, the angle at the center is

$\Delta\theta$

But when we draw $OM$ , it divides the triangle into two equal right triangles.

Therefore,

$\angle AOM=\Delta\theta/2$

At the same time, the chord is divided into two equal pieces:

$AM=c/2$

So both halves appear in the sine equation:

$\sin(\Delta\theta/2)=(c/2)/R$

That is why the final formula contains

$\boxed{\Delta\theta/2}$

Q8. What happens for a unit circle?

If the radius is

$R=1$

then

$c=2(1)\sin(\Delta\theta/2)$

Therefore,

$\boxed{c=2\sin(\Delta\theta/2)}$

This is the formula shown in our earlier quarter-circle example.

Q9. Can we use this formula for our quarter-circle?

Yes.

Suppose a quarter-circle has radius $1$ .

The total angle is

$\pi/2$

Suppose we divide it into 4 equal pieces.

Then the angle of each piece is

$\Delta\theta=(\pi/2)/4=\pi/8$

The chord corresponding to each piece is therefore

$c=2\sin(\pi/16)$

Numerically,

$c\approx0.39018$

There are 4 chords, so their total length is

$4c=4(0.39018)$

giving

$\boxed{4c\approx1.5607}$

Q10. Is $1.5607$ the actual quarter-circle arc length?

No.

This is a crucial distinction.

The actual quarter-circle arc length is

$\pi/2\approx1.5708$

The sum of the four chords is

$1.5607$

Therefore,

$1.5607<1.5708$

Why?

Because every chord is slightly shorter than its corresponding curved arc.

So:

$\boxed{\text{sum of chords}<\text{actual arc length}}$

Q11. Why does the chord approximation become better with more pieces?

Suppose we use 4 pieces.

The chords are relatively long, so the difference between a chord and its corresponding arc is noticeable.

Now use 100 pieces.

Each chord is much shorter and follows the curve much more closely.

Thus,

$\text{chord}\approx\text{small arc}$

As the number of pieces increases,

$n\rightarrow\infty$

the difference becomes smaller and smaller.

Conceptually:

$\boxed{\text{few pieces}\rightarrow\text{rough approximation}}$ $\boxed{\text{many pieces}\rightarrow\text{excellent approximation}}$

Q12. How does this eventually lead to arc length?

This is where our chord formula connects to calculus.

For each small section we calculate the chord:

$c_i=2R\sin(\Delta\theta_i/2)$

Then we add them:

$L_{\text{approx}}=c_1+c_2+c_3+\cdots+c_n$

Or,

$\boxed{L_{\text{approx}}=\sum_{i=1}^{n}c_i}$

As the number of pieces becomes larger and larger,

$n\rightarrow\infty$

the chords become indistinguishable from the corresponding tiny arcs.

The sum approaches the true arc length.

Eventually, calculus expresses this limiting accumulation as

$\boxed{L=\int ds}$

Q13. What is the relationship between $c$ , $\Delta s$ , and $ds$ ?

This is where keeping the notation separate is especially useful.

$c$

The length of a finite straight chord:

$\boxed{c=2R\sin(\Delta\theta/2)}$

$\Delta s$

The length of the finite curved arc between the two points:

$\boxed{\Delta s=\text{arc length between A and B}}$

For a finite piece,

$\boxed{c<\Delta s}$

$ds$

When that arc becomes infinitesimally small, we write its length as

$\boxed{ds}$

In the limiting sense,

$c\rightarrow ds$

So the conceptual progression is:

$\boxed{\text{finite arc }\Delta s}$

approximated by

$\boxed{\text{chord }c}$

and as the pieces become infinitesimally small,

$\boxed{c\rightarrow ds}$

Then all the tiny arc lengths are accumulated:

$\boxed{L=\int ds}$

The Big Picture

The entire argument can now be summarized in one chain:

$\boxed{\text{Circle geometry}}$ $\downarrow$ $\boxed{c=2R\sin(\Delta\theta/2)}$ $\downarrow$ $\boxed{\text{Add many chords}}$ $\downarrow$ $\boxed{\text{Make the pieces smaller}}$ $\downarrow$ $\boxed{c\rightarrow ds}$ $\downarrow$ $\boxed{L=\int ds}$

So the chord formula itself is simply a geometrical stepping stone. It allows us to understand how a curved length can be approximated by adding straight lengths, which eventually leads naturally to the calculus concept of arc length.

Chord Length Calculator

Calculate the straight-line length of a chord in a circle.

Chord length

c = 2r sin(θ/2)

For the angle method, the angle may be entered in degrees or radians. For the arc-length method, the arc length must correspond to the chosen radius.

Understanding the Chord Length Formula

How simple geometry gives us the formula for a chord

Q1. What is a chord?

Q2. What information do we need to find the chord?

Q3. What kind of triangle do we have?

Q4. What does the right triangle $OAM$ look like?

Q5. Why do we use sine?

Q6. How do we solve this for the chord length?

Q7. Why is there a $\Delta\theta/2$ rather than $\Delta\theta$ ?

Q8. What happens for a unit circle?

Q9. Can we use this formula for our quarter-circle?

Q10. Is $1.5607$ the actual quarter-circle arc length?

Q11. Why does the chord approximation become better with more pieces?

Q12. How does this eventually lead to arc length?

Q13. What is the relationship between $c$ , $\Delta s$ , and $ds$ ?

$c$

$\Delta s$

$ds$

The Big Picture

Chord Length Calculator

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DavidsonNext: AP® Calculus: Challenging Concepts from Calculus AB & Calculus BC

Calculus 1A: Differentiation

Calculus 1B: Integration

Calculus 1C: Coordinate Systems & Infinite Series

How simple geometry gives us the formula for a chord

Q1. What is a chord?

Q2. What information do we need to find the chord?

Q3. What kind of triangle do we have?

Q4. What does the right triangle look like?

Q5. Why do we use sine?

Q6. How do we solve this for the chord length?

Q7. Why is there a rather than ?

Q8. What happens for a unit circle?

Q9. Can we use this formula for our quarter-circle?

Q10. Is the actual quarter-circle arc length?

Q11. Why does the chord approximation become better with more pieces?

Q12. How does this eventually lead to arc length?

Q13. What is the relationship between , , and ?

The Big Picture

Chord Length Calculator

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Like this:

DavidsonNext: AP® Calculus: Challenging Concepts from Calculus AB & Calculus BC

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Calculus 1A: Differentiation

Calculus 1B: Integration

Calculus 1C: Coordinate Systems & Infinite Series

Q4. What does the right triangle $OAM$ look like?

Q7. Why is there a $\Delta\theta/2$ rather than $\Delta\theta$ ?

Q10. Is $1.5607$ the actual quarter-circle arc length?

Q13. What is the relationship between $c$ , $\Delta s$ , and $ds$ ?